Dispersive effects for the Schrödinger equation on finite metric graphs with infinite ends

We study the free Schrödinger equation on finite metric graphs with infinite ends. We give sufficient conditions to obtain the $L^1$ to $L^{\mathrm{\infty }}$ time decay rate at least $t^{-1\mathrm{/}2}$. These conditions allow certain metric graphs with circles and/or with commensurable lengths of the bounded edges. Further we study the dynamics of the probability flow between the bounded sub-graph and the unbounded ends.